Key Concepts and Formulas

• The implication $A \rightarrow B$ is false if and only if $A$ is true and $B$ is false.
• The conjunction $A \wedge B$ is true if and only if both $A$ and $B$ are true.
• The disjunction $A \vee B$ is true if at least one of $A$ or $B$ is true.
• The negation $\sim A$ is true if $A$ is false, and false if $A$ is true.

Step-by-Step Solution

Step 1: Analyze the given statement and its truth value.

We are given that $(P \wedge (\sim R)) \rightarrow ((\sim R) \wedge Q)$ is false. Using the fact that $A \rightarrow B$ is false only when $A$ is true and $B$ is false, we can deduce: $P \wedge (\sim R) \text{ is true}$ $(\sim R) \wedge Q \text{ is false}$

Step 2: Determine the truth values of P, Q, and R.

Since $P \wedge (\sim R)$ is true, both $P$ and $\sim R$ must be true. Thus, $P \text{ is true (T)}$ $\sim R \text{ is true (T)}$ This implies that $R \text{ is false (F)}$

Now, consider $(\sim R) \wedge Q$ is false. Since we know that $\sim R$ is true, for the conjunction to be false, $Q$ must be false. $Q \text{ is false (F)}$

So, we have $P = T$, $Q = F$, and $R = F$.

Step 3: Evaluate the truth values of the given options.

(A) $P \vee Q \rightarrow \sim R$ $P \vee Q$ is $T \vee F$, which is $T$. $\sim R$ is $\sim F$, which is $T$. Therefore, $P \vee Q \rightarrow \sim R$ is $T \rightarrow T$, which is $T$.

(B) $R \vee Q \rightarrow \sim P$ $R \vee Q$ is $F \vee F$, which is $F$. $\sim P$ is $\sim T$, which is $F$. Therefore, $R \vee Q \rightarrow \sim P$ is $F \rightarrow F$, which is $T$.

(C) $\sim (P \vee Q) \rightarrow \sim R$ $P \vee Q$ is $T \vee F$, which is $T$. $\sim (P \vee Q)$ is $\sim T$, which is $F$. $\sim R$ is $\sim F$, which is $T$. Therefore, $\sim (P \vee Q) \rightarrow \sim R$ is $F \rightarrow T$, which is $T$.

(D) $\sim (R \vee Q) \rightarrow \sim P$ $R \vee Q$ is $F \vee F$, which is $F$. $\sim (R \vee Q)$ is $\sim F$, which is $T$. $\sim P$ is $\sim T$, which is $F$. Therefore, $\sim (R \vee Q) \rightarrow \sim P$ is $T \rightarrow F$, which is $F$.

Step 4: Identify the option with a truth value of F.

From the evaluations above, we see that only option (D) has a truth value of F. However, the correct answer given is (A). There appears to be an error in the provided correct answer. We need to re-evaluate to find the error.

The original question states that we want to find the option that is FALSE.

Let's re-examine each option: (A) $P \vee Q \rightarrow \sim R$ $P=T, Q=F, R=F$ $(T \vee F) \rightarrow (\sim F)$ $T \rightarrow T$ which is $T$.

(B) $R \vee Q \rightarrow \sim P$ $F \vee F \rightarrow \sim T$ $F \rightarrow F$ which is $T$.

(C) $\sim (P \vee Q) \rightarrow \sim R$ $\sim (T \vee F) \rightarrow \sim F$ $\sim (T) \rightarrow T$ $F \rightarrow T$ which is $T$.

(D) $\sim (R \vee Q) \rightarrow \sim P$ $\sim (F \vee F) \rightarrow \sim T$ $\sim (F) \rightarrow F$ $T \rightarrow F$ which is $F$.

So (D) is the only option that is false. However, the correct answer says (A). There must be an error in the problem or the provided correct answer. Let's double-check the initial assignments of T/F to P, Q, and R.

$(P \wedge (\sim R)) \rightarrow ((\sim R) \wedge Q)$ is False. So $P \wedge (\sim R)$ is True and $(\sim R) \wedge Q$ is False. $P$ is True and $\sim R$ is True, which means $R$ is False. Since $\sim R$ is True, then for $(\sim R) \wedge Q$ to be False, $Q$ must be False. So $P=T$, $Q=F$, $R=F$.

Let's check option A again. $P \vee Q \rightarrow \sim R$ $T \vee F \rightarrow \sim F$ $T \rightarrow T$ which is $T$. This is NOT False.

Since we have re-verified all the steps, there is an error in the given correct answer. The correct answer should be (D).

Common Mistakes & Tips

• Carefully analyze the truth table for implications and conjunctions. A single mistake here will propagate through the entire solution.
• Remember that $A \rightarrow B$ is only false when $A$ is true and $B$ is false. In all other cases, it is true.
• Double-check your truth value assignments for P, Q, and R.

Summary

We are given that $(P \wedge (\sim R)) \rightarrow ((\sim R) \wedge Q)$ is false. From this, we deduced that $P$ is true, $Q$ is false, and $R$ is false. Substituting these values into the given options, we found that option (D), $\sim (R \vee Q) \rightarrow \sim P$, is the only one with a truth value of F. The originally provided answer was incorrect.

Final Answer

The final answer is \boxed{D}, which corresponds to option (D).